Vortical and Self-similar Flows of 2D Compressible Euler Equations

0.15 MB·English

by  Manwai Yuen

Vortical and Self-similar Flows of 2D
Compressible Euler Equations
Manwai Yuen∗
Department of Mathematics and Information Technology,
3 The Hong Kong Institute of Education,
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0 10 Po Ling Road, Tai Po, New Territories, Hong Kong
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n Revised 19-Jan-2013
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Abstract

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Thispaperpresentsthevorticalandself-similar solutions for2D compressibleEulerequa-
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- tions using the separation method. These solutions complement Makino’s solutions in radial
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t symmetry without rotation. The rotational solutions provide new information that furthers
a
m
our understanding of ocean vortices and reference examples for numerical methods. In ad-
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dition, the corresponding blowup, time-periodic or global existence conditions are classified

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through an analysis of the new Emden equation. A conjecture regarding rotational solutions
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9 in 3D is also made.
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MSC: 76U05, 35C05, 35C06, 35B10, 35R35
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3 KeyWords: Vortex,CompressibleEulerEquations,VorticalSolution,Self-similarSolution,
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: Time-periodic Solution
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r 1 Introduction
a
In fluid mechanics, the N-dimensional isentropic compressible Euler and Navier-Stokes equations
are expressed as follows:
ρ +∇·(ρ~u)=0
t
(1)

ρ[~ut+(~u·∇)~u]+K∇ργ =0,
where ρ = ρ(t,~x) denotes the density of the fluid, ~u = ~u(t,~x) = (u1,u2,··· ,uN) ∈ RN is the
velocity, ~x=(x ,x ,··· ,x )∈RN is the space variable, and K >0, γ >1 are constants.
1 2 N
∗E-mailaddress: [email protected]
1
2 M.W.Yuen
The Euler equations have been studied in great detail by numerous scholars because of their
significance in a variety of physical fields, such as fluids, plasmas, condensed matter, astrophysics,
oceanography, and atmospheric dynamics. These equations are also important in physics and
are widely used in different areas of study. For instance, the Euler system is the basic model
for shallow water flows [3]. It also provides a good model of the superfluids produced by the
Bose-Einstein condensates in the dilute gases of alkali metal atoms, in which identical gases do
not interact at very low temperatures [4]. At the microscopic level, fluids or gases are formed by
many tiny discrete molecules or particles that collide with one another. As the cost of directly
calculating the particle-to-particle or molecule-to-molecule evolution of fluids on a large scale is
expensive, approximation methods are needed to considerably simplify the process. Therefore, at
the macroscopic scale, the continuum assumption, which considers fluids as continuous, is applied
tothemodeling. Here,theEulerequationsprovideagooddescriptionofthefluidsatthestatistical
limit of a large number of small ideal molecules or particles by ignoring the less influential effects,
such as the self-gravitational forces and relativistic effect [1]. For a mathematical introduction of
the Euler equations, readers are referred to [8] and [2].
The constructionof analyticalor exact solutions is animportant area in mathematical physics
andapplied mathematics,as it canfurther classify their nonlinearphenomena. Fornon-rotational
flows,MakinofirstobtainedtheradialsymmetrysolutionsfortheEulerequations(1)inRN in1993
[9]. A number ofspecial solutionsfor these equations[6], [7], [12], [13], and [14]were subsequently
obtained. For rotational flows, Zhang and Zheng [15] constructed explicitly rotational solutions
for the Euler equations with γ =2 in 1997:
r2 1 1
ρ= , u = (x+y), u = (x−y), (2)
8Kt2 1 2t 2 2t
where x=rcosθ and y =rsinθ.
Very recently, Kwong and Yuen [5] constructed a family of rotational solutions for the Euler-
Poisson equations
ρ +∇·(ρ~u)=0
t
ρ(~u +(~u·∇)~u)+K∇ρ=−ρ∇Φ (3)
 t
∆Φ(t,~x)=2πρ,
with N =2 and γ =1: 
·
ρ(t,~x)= 1 ef(r/a(t)), ~u(t,~x)=a(t)(x,y)+ ξ (−y,x)
a(t)2 a(t) a(t)2
 a¨(t)= −λ + ξ2 , a(0)=a >0, a˙(0)=a (4)
 f··(s)+1f·a((st))+2πa(etf)3(s)=2λ, f(00)=α, f·(0)=10,
s K K
with arbitrary constants ξ 6=0, a0, a1 and α.
The rotational case complements Yuen’s solutions without rotation (ξ = 0) [11]. In this paper,
Vortical Flows of 2D Euler Equations 3
basedon the foregoingdevelopment, we provide the correspondingvorticalflows for 2D compress-
ible Euler equations (1) with γ >1 in the following result.
Theorem 1 For γ > 1, there exists a family of vortical flows in radial symmetry for the com-
pressible Euler equations (1) in 2D,
max (−λ(γ−1)s+α)γ−11, 0
ρ(t,~x)= (cid:18) 2Kγ (cid:19)
 a(t)2
 ~u(t,~x)=aa·((tt))(x,y)+a(ξt)2(−y,x) (5)
a¨(t)− ξ2 − λ =0, a(0)=a >0, a˙(0)=a ,
a(t)3 a(t)2γ−1 0 1
with the self-similar variable s= x2+y2 and arbitrary constants ξ 6=0, a , a , and α.
a(t)2 0 1
Remark 2 This result complements Makino’s solutions in radial symmetry without rotation (ξ =
0). The vortical solutions (5) provide new information that may further our understanding of
oceans vortices and reference examples for numerical methods in computational physics.
Remark 3 Zhang and Zheng’s solution (1) for γ =2 is a special case in our solutions (5).
2 2D Vortical and Self-similar Flows
Here, we provide a lemma for the vortical flows in 2D for the mass equation (1) . This lemma
1
originates in Kwong and Yuen’s paper [5], which constructs periodic and spiral solutions for 2D
Euler-Poissonequations (3) with γ =1.
Lemma 4 ([5]) For the equation of the conservation of mass,
ρ +∇·(ρ~u)=0, (6)
t
there exist the following solutions:
f( r ) a˙(t) G(t,r)
a(t)
ρ(t,~x)=ρ(t,r)= , ~u(t,~x)= (x,y)+ (−y,x) (7)
a(t)2 a(t) r
with the radial r = x2+y2 and arbitrary functions f ≥0; G and a(t)>0∈C1.
Proof. The functiopnal structure
F(t,r) G(t,r)
ρ(t,~x)=ρ(t,r), ~u(t,~x)= (x,y)+ (−y,x) (8)
r r
with an arbitrary C1 function F(t,r), can be plugged intothe mass equation (6) toverify the result:
ρ +∇·(ρ~u) (9)
t
∂ Fx Gy ∂ Fy Gx
=ρ + ρ −ρ + ρ +ρ (10)
t
∂x r r ∂y r r
(cid:20) (cid:21) (cid:20) (cid:21)
4 M.W.Yuen
∂ Fx ∂ Fx ∂ Gy ∂ Gy
=ρ + ρ +ρ −( ρ) −ρ( )
t
∂x r ∂x r ∂x r ∂x r
(cid:18) (cid:19)
∂ Fy ∂ Fy ∂ Gx ∂ Gx
+ ρ +ρ +( ρ) +ρ( ) (11)
∂y r ∂y r ∂y r ∂y r
(cid:18) (cid:19)
xFx x x F x
=ρ +ρ +ρ F +ρ −ρFx
t rr r rr r r r3
xGy xy(cid:16) (cid:17) x yFy y y
−ρ −ρG +ρGy +ρ +ρ F
rr r rr r r3 rr r rr r
F y yGx y x (cid:16) y (cid:17)
+ρ −ρFy +ρ +ρ G −ρGx (12)
r r3 rr r rr r r3
(cid:16) (cid:17)
xFx x x F x
=ρ +ρ +ρ F +ρ −ρFx
t rr r rr r r r3
yFy y(cid:16) y (cid:17) F y
+ρ +ρ F +ρ −ρFy (13)
rr r rr r r r3
(cid:16) (cid:17)
1
=ρ +ρ F +ρF +ρF . (14)
t r r
r
Then, the self-similar structure is taken for the density function,
f( r )
a(t)
ρ(t,~x)=ρ(t,r)= (15)
a(t)2
and F(t,r)= a˙(t)r for velocity ~u to balance equation (14) [11]:
a(t)
∂ f( r ) ∂ f( r )a˙(t)r f( r )a˙(t) f( r )a˙(t)
a(t) a(t) a(t) a(t)
= + + + (16)
∂t a(t)2 ∂r a(t)2 a(t) a(t)2 a(t) a(t)2 a(t)
· · ·
−2a(t)f(r/a(t)) a(t)rf(r/a(t))
= −
a(t)3 a(t)4
· · · ·
f(r/a(t))a(t)r f(r/a(t))a(t) f(r/a(t))a(t)
+ + + (17)
a(t)3 a(t) a(t)2 a(t) a(t)2 a(t)
=0. (18)
The proof is completed.
The computational proof for Theorem 1 is as follows.
Proof of Theorem 1. The procedure for the proof for vortical fluids is similar to that for
non-vorticalfluids [9], [14]. It is clear that the following function
·
f(s) a(t) ξ
ρ(t,~x)= , ~u(t,~x)= (x,y)+ (−y,x), (19)
a(t)2 a(t) a(t)2
with an arbitrary C1 function f ≥ 0 of the self-similar variable s = x2+y2 = r2 and a unde-
a(t)2 a(t)2
termined C2 time-function a(t) > 0, satisfies Lemma 4 for the mass equation (1) . For the first
1
momentum equation (1) , we obtain:
21
∂ ∂u ∂u ∂
=ρ u +u 1 +u 1 +Kγργ−1 ρ (20)
1 1 2
∂t ∂x ∂y ∂x
(cid:18) (cid:19)
Vortical Flows of 2D Euler Equations 5
∂ ∂u ∂u f(s)γ−1 1 ∂
1 1
=ρ u +u +u +Kγ f(s) (21)
∂t 1 1 ∂x 2 ∂y a(t)2(γ−1)a(t)2∂x
(cid:18) (cid:19)
∂ ∂u ∂u f(s)γ−2 2x
=ρ u +u 1 +u 1 +Kγ f˙(s) (22)
∂t 1 1 ∂x 2 ∂y a(t)2(γ−1)a(t)2
(cid:20) (cid:21)
∂ a˙(t)x− ξ y + a˙(t)x− ξ y ∂ a˙(t)x− ξ y
∂t a(t) a(t)2 a(t) a(t)2 ∂x a(t) a(t)2
=ρ (23)
+ (cid:16) ξ x+ a˙(t)y (cid:17)∂ (cid:16)a˙(t)x− ξ y(cid:17)+K(cid:16)γf(s)γ−22xf˙(s(cid:17))
a(t)2 a(t) ∂y a(t) a(t)2 a(t)2γ
 (cid:16) (cid:17) (cid:16) (cid:17) 
a¨(t) − a˙(t)2 x+ 2ξa˙(t)y+ a˙(t)x− ξ y a˙(t)
a(t) a(t)2 a(t)3 a(t) a(t)2 a(t)
=ρ (24)
(cid:16)+ ξ x+(cid:17)a˙(t)y − ξ (cid:16)+Kγf(s)γ−22x(cid:17)f˙(s)
a(t)2 a(t) a(t)2 a(t)2γ
 (cid:16) a¨(t) ξ2(cid:17)(cid:16) (cid:17) f(s)γ−2 
=ρ − x+Kγ 2xf˙(s) (25)
a(t) a(t)4 a(t)2γ
(cid:20)(cid:18) (cid:19) (cid:21)
ρx
= λ+2Kγf(s)γ−2f˙(s) , (26)
a(t)2γ
h i
with the Emden equation
a¨(t)− ξ2 = λ
a(t)3 a(t)2γ−1 (27)

a(0)=a >0, a˙(0)=a ,
 0 1
with arbitrary constants ξ and λ>1.

To ensure that equation (26) is zero, the following ordinary differential equation is required:
λ+2Kγf(s)γ−2f˙(s)=0
(28)
 f(0)=α>0, f˙(0)=0.

The function f(s) in the foregoing ordinary differential equation can be solved exactly:
1
λ(γ−1) γ−1
f(s)= − s+α . (29)
2Kγ
(cid:18) (cid:19)
For the second momentum equation (1) , we also have:
22
∂ ∂u ∂u ∂
=ρ u +u 2 +u 2 +Kγργ−1 ρ (30)
2 1 2
∂t ∂x ∂y ∂y
(cid:18) (cid:19)
∂ ∂u ∂u f(s)γ−2 2y
=ρ u +u 2 +u 2 +Kγ f˙(s) (31)
∂t 2 1 ∂x 2 ∂y a(t)2(γ−1)a(t)2
(cid:20) (cid:21)
∂ ξ x+ a˙(t)y + a˙(t)x− ξ y ∂ ξ x+ a˙(t)y
∂t a(t)2 a(t) a(t) a(t)2 ∂x a(t)2 a(t)
=ρ (32)
+ (cid:16) ξ x+ a˙(t)y(cid:17)∂ (cid:16) ξ x+ a˙(t)y(cid:17)+K(cid:16)γf(s)γ−22yf˙(s(cid:17))
a(t)2 a(t) ∂y a(t)2 a(t) a(t)2γ
 (cid:16) (cid:17) (cid:16) (cid:17) 
−2ξa˙(t)x+ a¨(t) − a˙(t)2 y+ a˙(t)x− ξ y ξ
a(t)3 a(t) a(t)2 a(t) a(t)2 a(t)2
=ρ (33)
 + ξ (cid:16)x+ a˙(t)y a˙(cid:17)(t) +K(cid:16) γf(s)γ−22yf˙((cid:17)s) 
a(t)2 a(t) a(t) a(t)2γ
 (cid:16)a¨(t) ξ2 (cid:17) f(s)γ−2 
=ρ − y+Kγ 2yf˙(s) (34)
a(t) a(t)4 a(t)2γ
(cid:20)(cid:18) (cid:19) (cid:21)
ρy
= λ+2Kγf(s)γ−2f˙(s) (35)
a(t)2γ
h i
6 M.W.Yuen
=0. (36)
with the Emden equation (27) and function (29).
Therefore, for ξ 6= 0, we have the vortical and self-similar flows for the compressible Euler
equations (1) in 2D [9].
To ensure the non-negative density function, we require that
1
max −λ(γ−1)s+α γ−1 , 0
2Kγ
ρ(t,~x)= (cid:18)(cid:16) (cid:17) (cid:19). (37)
a(t)2
This completes the proof.
Lemma 5 For the Emden equation,
a¨(t)− ξ2 − λ =0
a(t)3 a(t)2γ−1 (38)

a(0)=a >0,a˙(0)=a ,
 0 1
with arbitrary constants ξ 6=0, λ and γ >1, the total energy is
a˙(t)2 ξ2 λ
E(t):= + + . (39)
2 2a(t)2 (2γ−2)a(t)2γ−2
We have the following.
(1) For 1<γ <2, if E(0)<0, the solution is time-periodic; otherwise, the solution is global.
(2) For γ =2,
(2a) with ξ2 ≥−λ, the solution is global;
(2b) with ξ2 <−λ, the solution blows up at a finite time if
−λ−ξ2
a < ; (40)
1
a
p 0
otherwise, the solution is global.
(3) For γ >2,
(3a) with λ≥0, the solution is global;
1
(3b) with λ<0 and a constant a = −λ 2γ−4,
Max ξ2
(3bI) and a ≥a , (cid:16) (cid:17)
0 Max
if E(0) ≤ F (a ) or E(0) > F (a ) with a ≥ 0, the solution is global; otherwise, the
pot Max pot Max 1
solution blows up at a finite time.
(3bII) and a <a ,
0 Max
if E(0)≥F (a ) with a >0, thesolution is global; otherwise, the solution blows upat a finite
pot Max 1
time.
Proof. For equation (38), we multiply a˙(t) and then integrate it, as follows:
a˙(t)2 ξ2 λ
+ + =E(t), (41)
2 2a(t)2 (2γ−2)a(t)2γ−2
Vortical Flows of 2D Euler Equations 7
with a constant E(0)= a21 + ξ2 + λ .
2 2a20 (2γ−2)a20γ−2
We define the kinetic energy as:
a˙(t)2
F := , (42)
kin
2
and the potential energy as:
ξ2 λ
F = + . (43)
pot 2a(t)2 (2γ−2)a(t)2γ−2
The total energy is conserved thusly:
dE(t) d
= (F +F )=0. (44)
kin pot
dt dt
The classical energy method for second-order autonomous ordinary differential equations (which
readerscanrefertopages793–798in[10]),canbeappliedtoanalyzethecorrespondingqualitative
properties of the Emden equation (38).
(1)For1<γ <2,thereexistsauniqueglobalminimumforthepotentialfunction(43)fora(t)>0,
and lim F (a(t))=+∞ and lim F (a(t))=0. If
pot pot
a(t)→0+ a(t)→+∞
a2 ξ2 λ
E(0)= 1 + + <0, (45)
2 2a2 (2γ−2)a2γ−2
0 0
the solution is time-periodic; otherwise, it is global.
(2)Forγ =2,theordinarydifferentialequation(38)degeneratesintotheclassicalEmdenequation
ξ2+λ
a¨(t)= . (46)
a(t)3
(2a) For ξ2 ≥−λ, the solution is global.
(2b) For ξ2 < −λ, the potential function (43) is strictly increasing with lim F (a(t)) = 0.
pot
a(t)→+∞
The solution blows up at a finite time, if
−λ−ξ2
a < ; (47)
1
a
p 0
otherwise, the solution is global.
(3) For γ >2,
(3a)withλ≥0,thepotentialfunction(43)isadecreasingfunctionofa(t)>0. Thus,thesolution
is global.
(3b) With λ<0,the potentialfunction (43) achievesa unique globalmaximum, F (a ), with
pot Max
1
a = −λ 2γ−4 for a(t)>0.
Max ξ2
(3bI) An(cid:16)d a (cid:17)≥a ,
0 Max
if E(0) ≤ F (a ) or E(0) > F (a ) with a ≥ 0, the solution is global; otherwise, the
pot Max pot Max 1
solution blows up at a finite time.
(3bII) and a <a ,
0 Max
8 M.W.Yuen
ifE(0)≥F (a )witha >0,thesolutionisglobal;otherwisethesolutionblowsupatafinite
pot Max 1
time.
The proof is now complete.
The foregoing lemma makes it easy to determine the blowup or global existence of the corre-
sponding solutions (5) for the compressible Euler equations (1) in 2D.
Corollary 6 The total energy is defined
a˙(t)2 ξ2 λ
E(t):= + + , (48)
2 2a(t)2 (2γ−2)a(t)2γ−2
for the Emden equation (5) .
3
(1) For 1<γ <2, if E(0)<0, solution (5) is time-periodic; otherwise, solution (5) is global.
(2) For γ =2,
(2aI) with ξ2 >−λ or ξ2 =−λ and a ≥0, solution (5) is global;
1
(2aII) with ξ2 =−λ and a <0, solution (5) blows up at T =−a /a .
1 1 0
(2b) with ξ2 <−λ, solution (5) blows up at a finite time if
−λ−ξ2
a < ; (49)
1
a
p 0
otherwise, solution (5) is global.
(3) For γ >2,
(3a) with λ≥0, solution (5) is global;
1
(3b) with λ<0 and a constant a = −λ 2γ−4,
Max ξ2
(3bI) and a ≥a , (cid:16) (cid:17)
0 Max
if E(0)≤F (a ) or E(0)>F (a ) with a ≥0, solution (5) is global; otherwise, solution
pot Max pot Max 1
(5) blows up at a finite time.
(3bII) and a <a ,
0 Max
if E(0)≥F (a ) with a >0, solution (5) is global; otherwise, solution (5) blows up at a finite
pot Max 1
time.
3 Conclusion and Discussion
This paper provides a class of self-similar vortical flows for the 2D compressible Euler equations.
The resultpresentedhereincomplements Makino’ssolutions in radialsymmetry without rotation.
In addition, the corresponding blowup or global existence conditions are classified by analyzing
the new Emden equation (5) . Solution (5) is also the solution of the compressible Navier-Stokes
3
equations in 2D:
ρ +∇·(ρ~u) =0
t
(50)

ρ[~ut+(~u·∇)~u]+K∇ργ =µ∆~u,

Vortical Flows of 2D Euler Equations 9
with a positive constant µ.
Based on the existence of the rotational and self-similar solutions (5) in 2D, it is natural to
conjecture that there exists a class of rotationalsolutions for the compressible Euler equations (1)
in 3D, that complements those in radial symmetry, [9], [14]:
ρ= f(s)
3
 kΠ=1ak (51)
ui = aa˙ii (xi−d∗i)+d˙∗i, for i=1,2,3,
where 
1
ξ(γ−1) γ−1
f(s)=max − s+α , 0 , (52)
2Kγ
(cid:18) (cid:19) !
with
3 x −d∗
d∗ =d∗ +td∗ , s= ( k k)2. (53)
i i0 i1 a
k
k=1
X
Here, ξ,d∗ ,d∗ , α≥0 are constants, and functions a =a (t):
i0 i1 i i
a¨ = ξ for i=1,2,3
i 3 γ−1
 ai(cid:18)kΠ=1ak(cid:19) (54)
ai(0)=ai0 >0, a˙i(0)=ai1,
with arbitrary constants a and a .
i0 i1
Further researchwill be carried out to investigate the possibility of these vorticalsolutions in 3D.
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